Abstract

In this paper, we prove the global well posedness and the decay estimates for a $${\mathbb {Q } }$$-tensor model of nematic liquid crystals in $$\mathbb {R}^N$$, $$N \ge 3$$. This system is a coupled system by the Navier–Stokes equations with a parabolic-type equation describing the evolution of the director fields $${\mathbb {Q } }$$. The proof is based on the maximal $$L_p$$–$$L_q$$ regularity and the $$L_p$$–$$L_q$$ decay estimates to the linearized problem.

This paper is devoted to proving the existence of time-periodic solutions of one-phase or two-phase problems for the Navier–Stokes equations with small periodic external forces when the reference domain is close to a ball. Since our problems are formulated in time-dependent unknown domains, the problems are reduced to quasilinear systems of parabolic equations with non-homogeneous boundary conditions or transmission conditions in fixed domains by using the so-called Hanzawa transform. We separate solutions into the stationary part and the oscillatory part. The linearized equations for the stationary part have eigen-value 0, which is avoided by changing the equations with the help of the necessary conditions for the existence of solutions to the original problems. To treat the oscillatory part, we establish the maximal L –L regularity theorem of the periodic solutions for the system of parabolic equations with non-homogeneous boundary conditions or transmission conditions, which is obtained by the systematic use of R-solvers developed in Shibata (Diff Int Eqns 27(3–4):313–368, 2014; On the R-bounded solution operators in the study of free boundary problem for the Navier–Stokes equations. In: Shibata Y, Suzuki Y (eds) Springer proceedings in mathematics & statistics, vol. 183, Mathematical Fluid Dynamics, Present and Future, Tokyo, Japan, November 2014, pp 203–285, 2016; Comm Pure Appl Anal 17(4): 1681–1721. https://doi.org/10.3934/cpaa.2018081, 2018; R boundedness, maximal regularity and free boundary problems for the Navier Stokes equations, Preprint 1905.12900v1 [math.AP] 30 May 2019) to the resolvent problem for the linearized equations and the transference theorem obtained in Eiter et al. (R-solvers and their application to periodic L estimates, Preprint in 2019) for the L boundedness of operator-valued Fourier multipliers. These approaches are the novelty of this paper. p q p p

In this paper, we are concerned with generally symmetric hyperbolic-parabolic systems with Korteweg-type dispersion. Referring to those classical efforts in Kawashima et al., we formulate new structural conditions for the Korteweg-type dispersion and develop the dissipative mechanism of "regularity-gain type." As an application, it is checked that several concrete model systems (e.g., the compressible Navier-Stokes(-Fourier)-Korteweg system) satisfy the general structural conditions. In addition, the optimality of our general theory on the dissipative structure is also verified by calculating the asymptotic expansions of eigenvalues.

` We consider the Navier-Stokes-Korteweg equations for a viscous compressible fluid with capillarity effect. Referring to those studies in the non-capillary case, the purpose of this paper is to investigate the dissipation effect of Korteweg tensor with the density-dependent capillarity kappa(g). It is observed by the pointwise estimate that the linear third-order capillarity behaves like the heat diffusion of density fluctuation, which allows to develop the L-p energy methods (independent of spectral analysis). As a result, the time-decay estimates of L-q-L-r type regarding this system can be established. The treatment of nonlinear capillarity depends mainly on new Besov product estimates and the elaborate use of Sobolev embeddings and interpolations. Our results can be also applied to the quantum Navier-Stokes system, since it is a special choice of capillarity kappa(g) = kappa/g. (C) 2021 Elsevier Masson SAS. All rights reserved.

In this paper, we consider some two phase problems of compressible and incompressible viscous fluids’ flow without surface tension under the assumption that the initial domain is a uniform W domain in R (N ≥ 2). We prove the local in the time unique existence theorem for our problem in the L in time and Lq in space framework with 2 < p < ∞ and N < q < ∞ under our assumption. In our proof, we first transform an unknown time-dependent domain into the initial domain by using the Lagrangian transformation. Secondly, we solve the problem by the contraction mapping theorem with the maximal L -L regularity of the generalized Stokes operator for the compressible and incompressible viscous fluids’ flow with the free boundary condition. The key step of our proof is to prove the existence of an R-bounded solution operator to resolve the corresponding linearized problem. The Weis operator-valued Fourier multiplier theorem with R-boundedness implies the generation of a continuous analytic semigroup and the maximal L -L regularity theorem. q p p q p q 2-1/q N

In this paper, we consider the motion of incompressible magnetohydrodynamics (MHD) with resistivity in a domain bounded by a free surface. An electromagnetic field generated by some currents in an external domain keeps an MHD flow in a bounded domain. On the free surface, free boundary conditions for MHD flow and transmission conditions for electromagnetic fields are imposed. We proved the local well-posedness in the general setting of domains from a mathematical point of view. The solutions are obtained in an anisotropic space H ((0, T), H ) ∩ L ((0, T), H ) for the velocity field and in an anisotropic space H ((0, T), L ) ∩ L ((0, T), H ) for the magnetic fields with 2 < p < ∞, N < q < ∞ and 2/p + N/q < 1. To prove our main result, we used the Lp-Lq maximal regularity theorem for the Stokes equations with free boundary conditions and for the magnetic field equations with transmission conditions, which have been obtained by Frolova and the second author. 1 1 3 1 2 p q p q p q p q

In this paper, we consider the compressible fluid model of Korteweg type which can be used as a phase transition model. It is shown that the system admits a unique, global strong solution for small initial data in R , 3 ≤ N ≤ 7. In this study, the main tools are the maximal Lp-Lq regularity and Lp-Lq decay properties of solutions to the linearized equations. N

We show the existence of solution in the maximal L −L regularity framework to a class of symmetric parabolic problems on a uniformly C domain in R . Our approach consist in showing R - boundedness of families of solution operators to corresponding resolvent problems first in the whole space, then in half-space, perturbed half-space and finally, using localization arguments, on the domain. Assuming additionally boundedness of the domain we also show exponential decay of the solution. In particular, our approach does not require assuming a priori the uniform Lopatinskii - Shapiro condition. p q 2 n

In this paper, we establish some local and global solutions for the two-phase incompressible inhomogeneous flows with moving interfaces in the maximal Lp - Lq regularity class. Compared with previous results obtained by Solonnikov [Izv. Akad. Nauk SSSR Ser. Mat., 51 (1987), pp. 1065-1087, 1118 (in Russian); translation in Math. USSR-Isz., 31 (1988), pp. 381-405] and by Shibata and Shimizu [Differential Integral Equations, 20 (2007), pp. 241-276], we find the local solutions in the Lp - Lq class in some general uniform W - 1 domain in R by assuming (p, q) ∊]2, ∞[×]N, ∞[or (p, q) ∊]1, 2[×]N, ∞[satisfying 1/p + N/q > 3/2. In particular, the initial data with less regularity are allowed by assuming p < 2. In addition, if the density and the viscosity coefficient are piecewise constant, we can construct the long time solution from the small initial states in the case of the bounded droplet. This is due to some decay property for the corresponding linearized problem. r 2 /r N

In these lecture notes, we study free boundary problems for the Navier–Stokes equations with and without surface tension. The local well-posedness, global well-posedness, and asymptotics of solutions as time goes to infinity are studied in the L in time and L in space framework. To prove the local well-posedness, we use the tool of maximal L –L regularity for the Stokes equations with nonhomogeneous free boundary conditions. Our approach to proving maximal L –L regularity is based on the ℛ-bounded solution operators of the generalized resolvent problem for the Stokes equations with non-homogeneous free boundary conditions and the Weis operator-valued Fourier multiplier. Key to proving global well-posedness for the strong solutions is the decay properties of the Stokes semigroup, which are derived by spectral analysis of the Stokes operator in the bulk space and the Laplace–Beltrami operator on the boundary. We study the following two cases: (1) a bounded domain with surface tension and (2) an exterior domain without surface tension. In studying the latter case, since for unbounded domains we can obtain only polynomial decay in suitable L norms in space, to guarantee the L -integrability of solutions in time it is necessary to have the freedom to choose an exponent with respect to the time variable, thus it is essential to choose different exponents p and q. The basic approach of this chapter is to analyze the generalized resolvent problem, prove the existence of ℛ-bounded solution operators and determine the decay properties of solutions to the non-stationary problem. In particular, R-bounded solution operator and Weis’ operator valued Fourier multiplier theorem and transference theorem for the Fourier multiplier, we derive the maximal L –L regularity for the initial boundary value problem, find periodic solutions with non-homogeneous boundary conditions, and generate analytic semigroups for systems of parabolic equations, including equations appearing in fluid mechanics. This approach is quite new and extends the Fujita–Kato method in the study of the Navier–Stokes equations. p q p q p q q p p q

We consider the initial–boundary value problem for the system of equations describing the flow of compressible isothermal mixture of arbitrary large number of components. The system consists of the compressible Navier–Stokes equations and a subsystem of diffusion equations for the species. The subsystems are coupled by the form of the pressure and the strong cross-diffusion effects in the diffusion fluxes of the species. Assuming the existence of solutions to the symmetrized and linearized equations, proven in Piasecki, Shibata and Zatorska (2019), we derive the estimates for the nonlinear equations and prove the local-in-time existence and maximal L −L regularity of solutions. p q

In this paper, we prove the local and global well-posedness of some nonlinear thermoelastic plate equations with Dirichlet boundary conditions. The main tool for proving the local well-posedness is the maximal Lp-Lq regularity theorem for the linearized equations, and the main tool for proving the global well-posedness is the exponential stability of C0 analytic semigroup associated with linear thermoelastic plate equations with Dirichlet boundary conditions.

We consider the linear thermoelastic plate equations with free boundary conditions in uniform C4-domains, which includes the half-space, bounded and exterior domains. We show that the corresponding operator generates an analytic semigroup in Lp-spaces for all p∈(1,∞) and has maximal Lq-Lp-regularity on finite time intervals. On bounded C4 -domains, we obtain exponential stability.

This paper deals with the local well-posedness of free boundary problems for the Navier-Stokes equations in the case where the uid initially occupies an exterior domain Ω in N-dimensional Euclidian space ℝN.

In this chapter, we provide a review of results on the global well-posedness and optimal decay rate of strong solutions to the compressible Navier-Stokes equations in several type of domains: (1) whole space (Theorems 6, 7, 8, 9, 10, 11, and 12), (2) exterior domains (Theorems 13 and 14), (3) half-space (Theorem 15), (4) bounded domains (Theorem 16), and (5) infinite layers. Global well-posedness for the compressible viscous barotropic fluid motion with nonslip boundary condition was for the first time proved in the early 1980s by Matsumura and Nishida (Commun Math Phys 89:445-464, 1983) under the assumption that the H norm of the initial data is small. In Theorems 1, 2, 3, and 4, we revisit the same problem as in Matsumura and Nishida (Commun Math Phys 89:445- 464, 1983) under the weaker assumptions, namely, that the H norm of initial data is small. This is an improvement of the result in Matsumura and Nishida (Commun Math Phys 89:445-464, 1983) in view of the regularity assumption of the initial data. To show the methods, we perform the proof of Theorems 1, 2, 3, and 4 in all essential details. In this process, the L -L decay properties of solutions to the linearized equations are proved by using the cutoff technique and combining the local energy decay and the result in the whole space. This result was first proved by Kobayashi and Shibata (Commun Math Phys 200:621-659, 1999) under some additional assumption, and in this chapter, this assumption is eliminated by using a bootstrap argument. In the final section of this chapter, the optimal decay rate of the H norm of solution of the nonlinear problem is proved by combining the L -L decay properties of the linearized equations with some energy inequality of exponential decay type under the assumption that the initial data belong to the intersection space of H2 and L1. The main idea of this part of the proof is to combine the L -L decay properties of the Stokes semigroup and some Lyapunov-type energy inequality. 3 2 2 p q p q p q

In this chapter we present the classical energy approach for existence of regular solutions to the equations of compressible, heat-conducting fluids in a bounded three-dimensional domain. Firstly, we provide a state of the art and recall representative results in this field. Next, we give a proof of one of them, concerning Dirichlet boundary conditions for velocity and temperature. The result and thus the proof is divided into two main parts. A local-in-time existence result in high-regularity norms, via a method of successive approximations, occupies the former one. In the latter part, a differential inequality is derived, which allows us to extend the local-in-time solution to the global-in-time solution, provided a certain smallness condition is satisfied. This smallness condition is in fact an equilibrium proximity condition, since it involves differences between data and constants, whereas the data for temperature and density may be large themselves. All our considerations are performed within the L -approach. The proved result is close to that of Valli and Zajaczkowski (Commun Math Phys 103:259-296, 1986), but the techniques used here: the method of successive approximations (instead of a Leray-Schauder fixed-point argument there) as well as a clear continuation argument renders our exposition more traceable. Moreover, one may easily derive now an explicit smallness condition via our approach. Besides, the thermodynamic restriction on viscosities is relaxed, certain technicalities are improved and a possibly useful approach to deal with certain difficulties at the boundary in similar problems is provided. 2

This paper deals with the Lp-Lq decay estimate of the C0 analytic semigroup {T (t)}t≥0 associated with the perturbed Stokes equations with free boundary conditions in an exterior domain. The problem arises in the study of free boundary problem for the Navier-Stokes equations in an exterior domain. We proved that ||δjT(t)f||L p ≤ Cp,q t - j/2 - N/2(1/q - 1/p) ||f||L q (j = 0,1) provided that 1 &lt
q ≤ p ≤ ∞ and q ≠ ∞. Compared with the non-slip boundary condition case, the gradient estimate is better, which is important for the application to proving global well-posedness of free boundary problem for the Navier-Stokes equations. In our proof, it is crucial to prove the uniform estimate of the resolvent operator, the resolvent parameter ranging near zero.

In this paper, we prove the global well-posedness of free boundary problems of the Navier-Stokes equations in a bounded domain with surface tension. The velocity field is obtained in the Lp in time Lq in space maximal regularity class, (2&lt
p&lt
∞, N&lt
q&lt
∞, and 2/p+N/q&lt
1), under the assumption that the initial domain is close to a ball and initial data are sufficiently small. The essential point of our approach is to drive the exponential decay theorem in the Lp-Lq framework for the linearized equations with the help of maximal Lp-Lq regularity theory for the Stokes equations with free boundary conditions and spectral analysis of the Stokes operator and the Laplace-Beltrami operator.

We consider the motion of a viscous incompressible liquid crystal flow in the N -dimensional whole space. We prove the global well-posedness of strong solutions for small initial data by combining the maximal L-p - L-q regularities and L-p - L-q decay properties of solutions for the Stokes equations and heat equations. As a result, we also proved the decay properties of the solutions to the nonlinear equations.

We consider the linear thermoelastic plate equations with free boundary conditions in the L-p in time and L-q in space setting. We obtain unique solvability with optimal regularity for the inhomogeneous problem in a uniform C-4-domain, which includes the cases of a bounded domain and of an exterior domain with C-4-boundary. Moreover, we prove uniform a priori estimates for the solution. The proof is based on the existence of R-bounded solution operators of the corresponding generalized resolvent problem which is shown with the help of an operator-valued Fourier multiplier theorem due to Weis.

In this paper, we proved decay properties of solutions to the Stokes equations with surface tension and gravity in the half space R-+(N) = {(x', x(N)) x' is an element of RN-1, x(N) &gt; 0} (N &gt;= 2). In order to prove the decay properties, we first show that the zero points lambda(+/-) of Lopatinskii determinant for some resolvent problem associated with the Stokes equations have the asymptotics: lambda(+/-) = +/- ic(g)(1/2)vertical bar xi'vertical bar(1/2) - 2 vertical bar xi'vertical bar(2) + O(vertical bar xi'vertical bar(5/2)) as vertical bar xi'vertical bar -&gt; 0, where c(g) &gt; 0 is the gravitational acceleration and xi' is an element of RN-1 is the tangential variable in the Fourier space. We next shift the integral path in the representation formula of the Stokes semi-group to the complex left half-plane by Cauchy's integral theorem, and then it is decomposed into closed curves enclosing lambda(+/-) and the remainder part. We finally see, by the residue theorem, that the low frequency part of the solution to the Stokes equations behaves like the convolution of the (N - 1)-dimensional heat kernel and F-xi'(-1)[e(+/- icg1/2 vertical bar xi'vertical bar 1/2t)](x') formally, where F-xi'(-1) is the inverse Fourier transform with respect to xi'. However, main task in our approach is to show that the remainder part in the above decomposition decay faster than the residue part.

In this paper, we proved decay properties of solutions to the Stokes equations with surface tension and gravity in the half space R-+(N) = {(x', x(N)) x' is an element of RN-1, x(N) &gt; 0} (N &gt;= 2). In order to prove the decay properties, we first show that the zero points lambda(+/-) of Lopatinskii determinant for some resolvent problem associated with the Stokes equations have the asymptotics: lambda(+/-) = +/- ic(g)(1/2)vertical bar xi'vertical bar(1/2) - 2 vertical bar xi'vertical bar(2) + O(vertical bar xi'vertical bar(5/2)) as vertical bar xi'vertical bar -&gt; 0, where c(g) &gt; 0 is the gravitational acceleration and xi' is an element of RN-1 is the tangential variable in the Fourier space. We next shift the integral path in the representation formula of the Stokes semi-group to the complex left half-plane by Cauchy's integral theorem, and then it is decomposed into closed curves enclosing lambda(+/-) and the remainder part. We finally see, by the residue theorem, that the low frequency part of the solution to the Stokes equations behaves like the convolution of the (N - 1)-dimensional heat kernel and F-xi'(-1)[e(+/- icg1/2 vertical bar xi'vertical bar 1/2t)](x') formally, where F-xi'(-1) is the inverse Fourier transform with respect to xi'. However, main task in our approach is to show that the remainder part in the above decomposition decay faster than the residue part.

In this paper, we prove the maximal L-p-L-q regularity of the compressible and incompressible two phase flow with phase transition in the model problem case with the help of SI-bounded solution operators corresponding to generalized resolvent problem. The problem arises from the mathematical study of the motion, of two-phase flows having gaseous phase and liquid phase separated by a sharp interface with phase transition.

In this paper, we prove a local in time unique existence theorem for some two phase problem of compressible and compressible barotropic viscous fluid flow without surface tension in the L-p in time and the L-q in space framework with 2 &lt; p &lt; 1 and N &lt; q &lt; infinity under the assumption that the initial domain is a uniform W-q(2-1/q) domain in R-N (N &gt;= 2). After transforming a unknown time dependent domain to the initial domain by the Lagrangian transformation, we solve the problem by the contraction mapping principle with the maximal L-p-L-q regularity of the generalized Stokes operator for the compressible viscous fluid flow with free boundary condition. The key step of our method is to prove the existence of R-bounded solution operator to resolvent problem corresponding to linearized problem. The R-boundedness combined with Weis's operator valued Fourier multiplier theorem implies the generation of analytic semigroup and the maximal L-p-L-q regularity theorem.

In this paper, we prove a global in time unique existence theorem for the compressible viscous fluids in a bounded domain with slip boundary condition in the maximal L-p-L-q regularity class with 2 &lt; p &lt; infinity and N &lt; q &lt; infinity under the assumption that initial data are small enough and orthogonal to rigid motions if domain is rotationally symmetric, To prove the global well-posedness, we use the prolongation argument based on the maximal L-p-L-q regularity estimate of exponentially decay type. The same problem was first treated by Kobayashi and Zajaczkowski [5] in the L-2 framework by using the energy method; our approach is completely different from Kobayashi and Zajaczkowski [5]. (C) 2015 Elsevier Inc. All rights reserved.

In this paper, we prove the local well-posedness of the free boundary problems of Navier-Stokes equations in a general domain Omega subset of R-N (N &gt;= 2). The velocity field is obtained in the maximal regularity class W-q,p(2,1)(Omega x (0,T)) = L-p((0,T),W-q(2) (Omega) (N)) boolean AND W-p(1)((0,T); L-q (Omega) (N)) (2 &lt; p &lt; infinity and N &lt; q &lt; infinity) for any initial data satisfying certain compatibility conditions. The assumption of the domain Omega is the unique existence of solutions to the weak Dirichlet-Neumann problem as well as some uniformity of covering of the closure of Omega. A bounded domain, a perturbed half space, and a perturbed layer satisfy the conditions for the domain, and therefore drop problems and ocean problems are treated in the uniform manner. Our method is based on the maximal L-p-L-q regularity theorem of a linearized problem in a general domain.

In this paper, we prove a global in time unique existence theorem for the free boundary problem of a compressible barotropic viscous fluid flow without surface tension in the L-p in time and L-q in space framework with 2 &lt; p &lt; infinity and N &lt; q &lt; infinity under the assumption that the initial domain is bounded and initial data are small enough and orthogonal to rigid motions. Such global well-posedness was proved by Zajaczkowski in 1993 in the L-2 framework, and our result is an extension of his result to the maximal L-p-L-q regularity setting. We use the maximal L-p-L-q regularity theorem for the linearlized equations and the exponential stability of the corresponding analytic semigroup, which is a completely different approach than Zajaczkowski (1993).

In this paper, we consider the boundary value problem of Stokes operator arising in the study of free boundary problem for the Navier-Stokes equations with surface tension in a uniform W3−1/r r domain of N-dimensional Euclidean space ℝN (N ⩾ 2, N &lt
r &lt
∞). We prove the existence of R-bounded solution operator with spectral parameter λ varying in a sector Σε,λ0 = {λ ∈ ℂ | | arg λ| ⩽ π − ε, |λ| ⩾ λ0} (0 &lt
ε &lt
π/2), and the maximal Lp-Lq regularity with the help of the R-bounded solution operator and the Weis operator valued Fourier multiplier theorem. The essential assumption of this paper is the unique solvability of the weak Dirichlet-Neumann problem, namely it is assumed the unique existence of solution p ∈ W1 q (Ω) to the variational problem: (∇p,∇ϕ)Ω = (f,∇ϕ)Ω for any ϕ ∈ W1 q′(Ω) with 1 &lt
q &lt
∞ and q′ = q/(q − 1), where W1 q (Ω) is a closed subspace of Ŵ1 q,Γ (Ω) = {p ∈ Lq,loc(Ω) | ∇p ∈ Lq(Ω)N, p|Γ = 0} with respect to gradient norm ∥∇ · ∥Lq( Ω) that contains a space W1 q, Γ (Ω) = {p ∈ W1 q (Ω) | p|Γ = 0}, and Γ is one part of boundary on which free boundary condition is imposed. The unique solvability of such weak Dirichlet-Neumann problem is necessary for the unique existence of a solution to the resolvent problem with uniform estimate with respect to spectral parameter varying in (λ0,∞), which was proved in Shibata [13]. Our assumption is satisfied for any q ∈ (1,∞) by the following domains: half space, perturbed half space, bounded domains, layer, perturbed layer, straight cube, and exterior domains with W1 q (Ω) = Ŵ1 q,Γ (Ω).

This paper concerns the free boundary problem for the Navier-Stokes equations without surface tension in the Lp in time and Lq in space setting with 2&lt
p&lt
∞ and N&lt
q&lt
∞. A local in time existence theorem is proved in a uniform Wq2-1/q domain in the N-dimensional Euclidean space RN (N≥2) under the assumption that the weak Dirichlet-Neumann problem is uniquely solvable. Moreover, a global in time existence theorem is proved for small initial data under the additional assumption that Ω is bounded. This was already proved by Solonnikov [25] by using the continuation argument of local in time solutions which are exponentially stable in the energy level under the assumption that the initial data is orthogonal to the rigid motion. We also use the continuation argument and the same orthogonality condition for the initial data. But, our argument about the continuation of local in time solutions is based on some decay theorem for the linearized problem, which is a different point than [25].

This paper concerns the free boundary problem for the Navier-Stokes equations without surface tension in the L-p in time and L-q in space setting with 2 &lt; p &lt; infinity and N &lt; q &lt; infinity. A local in time existence theorem is proved in a uniform w(q)(2-1/q) domain in the N-dimensional Euclidean space R-N (N &gt;= 2) under the assumption that the weak Dirichlet Neumann problem is uniquely solvable. Moreover, a global in time existence theorem is proved for small initial data under the additional assumption that Omega is bounded. This was already proved by Solonnikov [25] by using the continuation argument of local in time solutions which are exponentially stable in the energy level under the assumption that the initial data is orthogonal to the rigid motion. We also use the continuation argument and the same orthogonality condition for the initial data. But, our argument about the continuation of local in time solutions is based on some decay theorem for the linearized problem, which is a different point than [25]. (C) 2015 Published by Elsevier Inc.

Given a uniform W-q(3-1/q)-domain Omega subset of R-N for N &lt; q &lt; infinity, we consider a simplified Ericksen-Leslie system modeling the flow of compressible nematic liquid crystals based on Lin and Liu [Comm. Pure Appl. Math., 48 (1995), pp. 501-537]. We show the unique existence of local-intime strong solutions. Furthermore, if Omega is bounded and initial data are chosen suitably small, we obtain global-in-time strong solutions. Our approach is based on maximal regularity estimates of the compressible Navier-Stokes equation by Enomoto, von Below, and Shibata [Ann. Univ. Ferrara, 60 (2014), pp. 55-89] and maximal regularity estimates for the Neumann problem as a consequence of Weis's 2001 vector-valued Fourier multiplier theorem.

The situation of this paper is that the Stokes equation for the compressible viscous fluid flow in the upper half-space is coupled via inhomogeneous interface conditions with the Stokes equations for the incompressible one in the lower half-space, which is the model problem for the evolution of compressible and incompressible viscous fluid flows with a sharp interface. We show the existence of R-bounded solution operators to the corresponding generalized resolvent problem, which implies the generation of analytic semigroup and maximal Lp-Lq regularity for the corresponding time dependent problem with the help of the Weis' operator valued Fourier multiplier theorem. The problem was studied by Denisova (Interfaces Free Bound. 2(3):283-312, 2000) under some restriction on the viscosity coefficients and one of our purposes is to eliminate the assumption in (Denisova in Interfaces Free Bound. 2(3):283-312, 2000).

The situation of this paper is that the Stokes equation for the compressible viscous fluid flow in the upper half-space is coupled via inhomogeneous interface conditions with the Stokes equations for the incompressible one in the lower half-space, which is the model problem for the evolution of compressible and incompressible viscous fluid flows with a sharp interface. We show the existence of R-bounded solution operators to the corresponding generalized resolvent problem, which implies the generation of analytic semigroup and maximal Lp-Lq regularity for the corresponding time dependent problem with the help of the Weis' operator valued Fourier multiplier theorem. The problem was studied by Denisova (Interfaces Free Bound. 2(3):283-312, 2000) under some restriction on the viscosity coefficients and one of our purposes is to eliminate the assumption in (Denisova in Interfaces Free Bound. 2(3):283-312, 2000).

In this paper, we consider the boundary value problem of Stokes operators with variable viscosity in the case of free boundary condition in a uniform W-r(2-1/r) domain of N-dimensional Euclidean space R-N (N &gt;= 2, N &lt; r &lt; infinity). We prove the R-boundedness of solution operators with spectral parameter A varying in a sector Sigma(epsilon),(lambda 0) = {lambda is an element of C : vertical bar arg lambda vertical bar &lt;= pi - epsilon, vertical bar lambda vertical bar &gt;= lambda(0)}, from which we can deduce the L-p-L-q maximal regularity as well as the generation of analytic semigroup for the time dependent problem. The problem of this type arises in the mathematical study of the incompressible viscous fluid flow with free surface. The essential assumption of this paper is the unique solvability of the weak Dirichlet-Neumann problem, namely, it is assumed the unique existence of solution p is an element of W-q(1)(Omega) to the variational problem: (del p, del phi)(Omega) = (f, del phi)(Omega) for any phi is an element of W-q(1), (Omega) with 1 &lt; q &lt; infinity and q' = q/(q - 1), where W-q(1)(Omega) is a closed subspace of (W) over cap (1)(q Gamma)(Omega) = {p is an element of L-q,(loc)(Omega) : del(p) is an element of L-q(Omega)(N), p vertical bar Gamma = 0} with respect to gradient norm parallel to del.parallel to(Lq(Omega)) that contains a space W-q,Gamma(1)(Omega) = {p is an element of W-q(1)(Omega) : p vertical bar Gamma = 0}, and Gamma is one part of boundary on which free boundary condition is imposed. The unique solvability of such weak Dirichlet-Neumann problem is necessary for the unique existence of a solution to the resolvent problem with uniform estimate with respect to spectral parameter varying in (lambda(0), infinity), which was proved in Shibata [28]. Our assumption is satisfied for any q is an element of (1, infinity) by the following domains: half space, perturbed half space, bounded domains, layer, perturbed layer with W-q(1)(Omega) = (W) over cap (1)(q,Gamma)(Omega), and by exterior domains with W-q,Gamma(1)(Omega) = the closure of W-q,Gamma(1)(Omega) with respect to the gradient norm. Combining the result in this paper with that in a forthcoming paper about the nonlinear problems, we can conclude that the unique existence of solutions to weak Dirichlet-Neumann problem implies a local in time unique existence theorem of strong solutions to the free boundary problem without surface tension taken into account for the Navier-Stokes equations in a uniform W-r(2-1/r) domain.

In this paper, we consider a generalized resolvent problem for the linearization system of the Navier-Stokes equations describing some free boundary problem of a compressible barotropic viscous fluid flow without taking the surface tension into account. We prove the existence of the R-bounded solution operators, which drives not only the generation of analytic semigroup but also the maximal L-p-L-q regularity by means of Weis' operator valued Fourier multiplier theorem for the corresponding time dependent problem that enable us to prove a local in time existence theorem of the free boundary problem for a compressible barotropic viscous fluid flow in the L-p in time and L-q space setting (cf. Annali dell Universita di Ferrara 60 (2014), 55-89). The results in this paper were given in the PhD thesis [Three topics in fluid dynamics: viscoelastic, generalized Newtonian, and compressible fluids, 2012, TU Darmstadt] by the first author under supervision of the second author. Here we present a slightly different method of deriving a concrete form of solutions to the model problem. In this paper, one of the essential points is to show the invertibility of a 2 x 2 Lopatinski matrix function. The corresponding system in [Three topics in fluid dynamics: viscoelastic, generalized Newtonian, and compressible fluids, 2012, TU Darmstadt] is a 3 x 3 matrix, so that the method presented here is slightly simpler.

In this paper, we consider a generalized resolvent problem for the linearization system of the Navier-Stokes equations describing some free boundary problem of a compressible barotropic viscous fluid flow without taking the surface tension into account. We prove the existence of the R-bounded solution operators, which drives not only the generation of analytic semigroup but also the maximal L-p-L-q regularity by means of Weis' operator valued Fourier multiplier theorem for the corresponding time dependent problem that enable us to prove a local in time existence theorem of the free boundary problem for a compressible barotropic viscous fluid flow in the L-p in time and L-q space setting (cf. Annali dell Universita di Ferrara 60 (2014), 55-89). The results in this paper were given in the PhD thesis [Three topics in fluid dynamics: viscoelastic, generalized Newtonian, and compressible fluids, 2012, TU Darmstadt] by the first author under supervision of the second author. Here we present a slightly different method of deriving a concrete form of solutions to the model problem. In this paper, one of the essential points is to show the invertibility of a 2 x 2 Lopatinski matrix function. The corresponding system in [Three topics in fluid dynamics: viscoelastic, generalized Newtonian, and compressible fluids, 2012, TU Darmstadt] is a 3 x 3 matrix, so that the method presented here is slightly simpler.

In this paper, we prove a local in time unique existence theorem for the freeboundary problem of a compressible barotropic viscous fluid flow without surface tensionin the Lp in time and Lq in space framework with 2 &lt
p &lt
∞ and N &lt
q &lt
∞ under the assumption that the initial domain is a uniform W2-1/qq one in ℝN (N ≥ 2).After transforming a unknown time dependent domain to the initial domain by theLagrangian transformation, we solve problem by the Banach contraction mappingprinciple based on the maximal Lp-Lq regularity of the generalized Stokes operatorfor the compressible viscous fluid flowwith free boundary condition. The key issue forthe linear theorem is the existence of R-bounded solution operator in a sector, whichcombined with Weis's operator valued Fourier multiplier theorem implies the generationof analytic semigroup and the maximal Lp-Lq regularity theorem. The nonlinearproblem we studied here was already investigated by several authors (Denisova andSolonnikov, St. Petersburg Math J 14:1-22, 2003
J Math Sci 115:2753-2765, 2003
Secchi, Commun PDE 1:185-204, 1990
Math Method Appl Sci 13:391-404, 1990
Secchi and Valli, J Reine Angew Math 341:1-31, 1983
Solonnikov and Tani, Constantincarathéodory: an international tribute, vols 1, 2, pp 1270-1303,World ScientificPublishing, Teaneck, 1991
Lecture notes in mathematics, vol 1530, Springer, Berlin,1992
Tani, J Math Kyoto Univ 21:839-859, 1981
Zajaczkowski, SIAM JMath Anal25:1-84, 1994) in the L2 framework and Hölder spaces, but our approach is differentfrom them. © 2013 Università degli Studi di Ferrara.

In this paper, we prove the R-sectoriality of the resolvent problem for the boundary value problem of the Stokes operator for the compressible viscous fluids in a general domain, which implies the generation of analytic semigroup and the maximal L-p-L-q regularity for the initial boundary value problem of the Stokes operator. Combining our linear theory with fixed point arguments in the Lagrangian coordinates, we have a local in time unique existence theorem in a general domain and a global in time unique existence theorem for some initial data close to a constant state in a bounded domain for the initial boundary value problem of the Navier-Stokes equations describing the motion of compressible viscous fluids. All the results obtained in this paper were announced in Enomoto-Shibata [12].

In this paper, we prove the R-sectoriality of the resolvent problem for the boundary value problem of the Stokes operator for the compressible viscous fluids in a general domain, which implies the generation of analytic semigroup and the maximal L-p-L-q regularity for the initial boundary value problem of the Stokes operator. Combining our linear theory with fixed point arguments in the Lagrangian coordinates, we have a local in time unique existence theorem in a general domain and a global in time unique existence theorem for some initial data close to a constant state in a bounded domain for the initial boundary value problem of the Navier-Stokes equations describing the motion of compressible viscous fluids. All the results obtained in this paper were announced in Enomoto-Shibata [12].

In this paper, we prove unique existence of solutions to the generalized resolvent problem of the Stokes operator with first order boundary condition in a general domain of the N-dimensional Eulidean space . This type of problem arises in the mathematical study of the flow of a viscous incompressible one-phase fluid with free surface. Moreover, we prove uniform estimates of solutions with respect to resolvent parameter varying in a sector , where and . The essential assumption of this paper is the existence of a unique solution to a suitable weak Dirichlet problem, namely it is assumed the unique existence of solution to the variational problem: for any . Here, is the closure of by the semi-norm , and is the boundary of . In fact, we show that the unique solvability of such a Dirichlet problem is necessary for the unique existence of a solution to the resolvent problem with uniform estimate with respect to resolvent parameter varying in . Our assumption is satisfied for any by the following domains: whole space, half space, layer, bounded domains, exterior domains, perturbed half space, perturbed layer, but for a general domain, we do not know any result about the unique existence of solutions to the weak Dirichlet problem except for q = 2.

In this paper, we proved the local energy decay and some L-p-L-q decay properties of solutions to the initial-boundary value problem for the Stokes equations of compressible viscous fluid flow in a 2-dimensional exterior domain. Kobayashi (1997) [19] and Kobayashi and Shibata (1999) [21] treated the same problem in a 3-dimensional exterior domain, and our results obtained in this paper are an extension of results in Kobayashi (1997) [19] and Kobayashi and Shibata (1999) [21] to the 2-dimensional case. The fundamental solution to the resolvent equations for the Stokes equations in R-2 has a logarithmical singularity at lambda = 0, lambda being a resolvent parameter, while it is continuous up to lambda = 0 in R-3. This difference requires us a new idea to prove the local energy decay estimate. Once getting the local energy decay estimate, the required L-p-L-q decay estimates in the exterior domain are obtained by combining the local energy estimate and the L-p-L-q estimates in R-2 by a cut-off technique. (C) 2012 Elsevier Inc. All rights reserved.

The basic model for incompressible two-phase flows with phase transitions is derived from basic principles and shown to be thermodynamically consistent in the sense that the total energy is conserved and the total entropy is nondecreasing. The local well-posedness of such problems is proved by means of the technique of maximal Lp-regularity in the case of equal densities. This way we obtain a local semiflow on a well-defined nonlinear state manifold. The equilibria of the system in absence of external forces are identified and it is shown that the negative total entropy is a strict Ljapunov functional for the system. If a solution does not develop singularities, it is proved that it exists globally in time, its orbit is relatively compact, and its limit set is nonempty and contained in the set of equilibria.

In this paper, we proved the local energy decay and some L-p-L-q decay properties of solutions to the initial-boundary value problem for the Stokes equations of compressible viscous fluid flow in a 2-dimensional exterior domain. Kobayashi (1997) [19] and Kobayashi and Shibata (1999) [21] treated the same problem in a 3-dimensional exterior domain, and our results obtained in this paper are an extension of results in Kobayashi (1997) [19] and Kobayashi and Shibata (1999) [21] to the 2-dimensional case. The fundamental solution to the resolvent equations for the Stokes equations in R-2 has a logarithmical singularity at lambda = 0, lambda being a resolvent parameter, while it is continuous up to lambda = 0 in R-3. This difference requires us a new idea to prove the local energy decay estimate. Once getting the local energy decay estimate, the required L-p-L-q decay estimates in the exterior domain are obtained by combining the local energy estimate and the L-p-L-q estimates in R-2 by a cut-off technique. (C) 2012 Elsevier Inc. All rights reserved.

The basic model for incompressible two-phase flows with phase transitions is derived from basic principles and shown to be thermodynamically consistent in the sense that the total energy is conserved and the total entropy is nondecreasing. The local well-posedness of such problems is proved by means of the technique of maximal Lp-regularity in the case of equal densities. This way we obtain a local semiflow on a well-defined nonlinear state manifold. The equilibria of the system in absence of external forces are identified and it is shown that the negative total entropy is a strict Ljapunov functional for the system. If a solution does not develop singularities, it is proved that it exists globally in time, its orbit is relatively compact, and its limit set is nonempty and contained in the set of equilibria.

In this paper, we proved the generalized resolvent estimate and the maximal L-p-L-q regularity of the Stokes equation with first order boundary condition in the half-space, which arises in the mathematical study of the motion of a viscous incompressible one phase fluid flow with free surface. The core of our approach is to prove the R boundedness of solution operators defined in a sector Sigma(epsilon,gamma 0) = {lambda is an element of C \ {0} vertical bar vertical bar arg lambda vertical bar &lt;= pi - epsilon, vertical bar lambda vertical bar &gt;= gamma(0)} with 0 &lt; epsilon &lt; pi/2 and gamma(0) &gt;= 0. This R boundedness implies the resolvent estimate of the Stokes operator and the combination of this R boundedness with the operator valued Fourier multiplier theorem of L. Weis implies the maximal L-p-L-q regularity of the non-stationary Stokes. For a densely defined closed operator A, we know that what A has maximal L-p regularity implies that the resolvent estimate of A in lambda is an element of Sigma(epsilon,gamma 0), but the opposite direction is not true in general (cf. Kalton and Lancien [19]). However, in this paper using the R boundedness of the operator family in the sector Sigma(epsilon,lambda 0), we derive a systematic way to prove the resolvent estimate and the maximal L-p regularity at the same time.

In this paper we consider the set of equations describing Oldroyd-B fluids in exterior domains. It is shown that these equations admit a unique, global solution defined in a certain function space provided the initial data and the coupling constant are small enough. (C) 2011 Elsevier Inc. All rights reserved.

In this paper we prove the generalized resolvent estimate and maximal L-p-L-q regularity of the Stokes equation with and without surface tension and gravity in the whole space with flat interface. We prove II boundedness of solution operators defined in a sector Sigma(epsilon,gamma 0) = {lambda is an element of C\ {0} vertical bar vertical bar arg lambda vertical bar &lt;= pi - epsilon vertical bar lambda vertical bar &gt;= gamma(0)] with 0 &lt; epsilon &lt; pi/2 and gamma(0) &gt;= 0, which combined with the Fourier multiplier theorem of S.G. Mihlin and the operator valued Fourier multiplier theorem of L Weis yields the required generalized resolvent estimate and maximal L-p-L-q regularity at the same time. One of the character of the paper is to introduce special function spaces E-q((R) over dot(n),Sigma(epsilon,gamma 0) and E-p,E-q,E-gamma 0 ((R) over dot(n) x R) (cf. (1.7) and (1.8)), which is necessary to treat the situation that the normal component of velocity fields jumps across the interface. Such spaces never appear in the study of the Stokes equations with other boundary conditions like non-slip condition, Navier slip condition, Robin condition or pure Neumann condition appearing in the study of one phase problem (cf. Desch et al., 2001 [12], Farwig and Sohr, 1994 [13], Saal, 2003 [22], Shibata and Shimada, 2007 [23], Shibata and Shimizu 2008 [25], 2009 [26], in press [27]), because the normal component of the velocity fields vanishes at the boundary which is physical requirement that the flow does not go out and come in through the rigid boundary. (C) 2011 Elsevier Inc. All rights reserved.

We consider the free boundary problem of the Navier-Stokes equation with surface tension. Our initial domain is one of a bounded domain, an exterior domain, a perturbed half-space or a perturbed layer in n (n epsilon 2). We report a local in time unique existence theorem in the space [image omitted] with some T 0, 2 p and n q for any initial data which satisfy compatibility condition. Our theorem can be proved by the standard fixed point argument based on the Lp-Lq maximal regularity theorem for the corresponding linearized equations. Our results cover the cases of a drop problem and an ocean problem that were studied by Solonnikov (Solvability of the evolution problem for an isolated mass of a viscous incompressible capillary liquid, Zap. Nauchn. Sem. (LOMI) 140 (1984) pp. 179-186 (in Russian) (English transl.: J. Soviet Math. 32 (1986), pp. 223-238)), Solonnikov (Unsteady motion of a finite mass of fluid, bounded by a free surface, Zap. Nauchn. Sem. (LOMI) 152 (1986), pp. 137-157 (in Russian) (English transl.: J. Soviet Math. 40 (1988), pp. 672-686)), Solonnikov (On nonstationary motion of a finite isolated mass of self-gravitating fluid, Algebra Anal. 1 (1989), pp. 207-249 (in Russian) (English transl.: Leningrad Math. J. 1 (1990), pp. 227-276)), Solonnikov (Solvability of the problem of evolution of a viscous incompressible fluid bounded by a free surface on a finite time interval, Algebra Anal. 3 (1991), pp. 222-257 (in Russian) (English transl.: St. Petersburg Math. J. 3 (1992) 189-220)), Beale (Large time regularity of viscous surface waves, Arch. Rat. Mech. Anal. 84 (1984), pp. 307-352) and Tani (Small-time existence for the three-dimensional incompressible Navier-Stokes equations with a free surface, Arch. Rat. Mech. Anal. 133 (1996), pp. 299-331).

We consider the initial-boundary value problem in hyperbolic thermoelasticity with second sound in a three-dimensional exterior domain. The low frequency expansion of solutions to the corresponding stationary resolvent problem is given and the limit to the classical thermoelastic problem is investigated.

We consider the initial-boundary value problem in hyperbolic thermoelasticity with second sound in a three-dimensional exterior domain. The low frequency expansion of solutions to the corresponding stationary resolvent problem is given and the limit to the classical thermoelastic problem is investigated.

Consider the Navier-Stokes equations in the rotational framework. It is proved that these equations possess a unique global mild solution for arbitrary speed of rotation provided the initial data u(0) is small enough in the H(sigma)(1/2)(R(3))-norm.

Consider the Navier-Stokes equations in the rotational framework. It is proved that these equations possess a unique global mild solution for arbitrary speed of rotation provided the initial data u(0) is small enough in the H(sigma)(1/2)(R(3))-norm.

In this paper we prove frequency expansions of the resolvent and local energy decay estimates for the linear thermoelastic plate equations:
u(tt) + Delta(2)u + Delta theta = 0 and theta(t) - Delta theta - Delta u(t) = 0 in Omega x (0, infinity).
subject to Dirichlet boundary conditions: u vertical bar Gamma = D(v)u vertical bar Gamma = theta vertical bar Gamma = 0 and initial conditions (u, u(t), theta)vertical bar(t=0) = (u(0), v(0), theta(0)) Here Omega is an exterior domain (domain with bounded complement) in R(n) with n = 2 or n = 3, the boundary F of which is assumed to be a C(4)-hypersurface.

In this paper we prove frequency expansions of the resolvent and local energy decay estimates for the linear thermoelastic plate equations:
u(tt) + Delta(2)u + Delta theta = 0 and theta(t) - Delta theta - Delta u(t) = 0 in Omega x (0, infinity).
subject to Dirichlet boundary conditions: u vertical bar Gamma = D(v)u vertical bar Gamma = theta vertical bar Gamma = 0 and initial conditions (u, u(t), theta)vertical bar(t=0) = (u(0), v(0), theta(0)) Here Omega is an exterior domain (domain with bounded complement) in R(n) with n = 2 or n = 3, the boundary F of which is assumed to be a C(4)-hypersurface.

In this paper, we show the unique existence of solutions to the nonstationary problem for the modified Oseen equation with rotating effect in Omega:
D(t)u-Delta u + kD(3)u + M(a)u + del p = 0, div u = 0 in Omega x (0, infinity),
(OS)
u vertical bar partial derivative Omega = 0, u vertical bar(t=0) = f,
where Omega is an exterior domain in R-3, M(a)u = -a(e(3) x x). del u + ae(3) x u, x = (x(1),x(2),x(3))is an element of R-3 and e(3) = (0, 0, 1). This problem arises from a linearization of the Navier Stokes equations describing an incompressible viscous fluid flow past a rotating obstacle. If 1 &lt; q &lt; infinity and initial data f satisfies the conditions: f is an element of W-q(2)(Omega), div f = 0 in Omega, f vertical bar partial derivative Omega = 0 and M(a)f is an element of L-q(Omega), then problem (OS) admits a unique solution (u(t), p(t)) which satisfies the following conditions:
u(t) is an element of C-1([0, infinity), L-q (Omega)) boolean AND C-0, ([0, infinity), w(Q)(2) (Omega)), p (t) is an element of C-0([0, infinity), (W) over cap (1)(q)(Omega)),
parallel to(u(t), t(1/2)del u(t), t del(2)u(t), del p(t))parallel to(Lq(Omega)) &lt;= C-a0,C-k0,C-gamma E-gamma t parallel to f parallel to(Lq(Omega)) ,
t((1/2)-(1+q)))parallel to-p(t) parallel to(Lq(Omega b)) &lt;= C-a0,E-k0,b,gamma(gamma t)parallel to f parallel to(Lq(Omega)),
parallel to D(t)u(t)parallel to(Lq(Omega)) + parallel to u(t)parallel to (Wq2(Omega)) + parallel to del p(t)parallel to(Lq(Omega)) &lt;=
C-a0,k(0,gamma)E(gamma t) (parallel to f parallel to(Wq2(Omega)) + parallel to M(a)f parallel to(Lq(Omega)))
for any t &gt; 0 and large positive gamma, where b is any number such that B-b superset of R-3 \ Omega and Omega(b) = B-b boolean AND Omega with B-b = {x is an element of R-3 vertical bar vertical bar x vertical bar &lt; b}. The estimate for pressure term p is new and important for further researches of the corresponding full nonlinear problem. We also prove the generation of a continuous semigroup associated with problem (OS), which has been announced in Shibata [20, Theorem 1.1]. The result

The paper is concerned with linear thermoelastic plate equations in the half-space R-+(n) = {x = (x(1),...,x(n)) | x(n) &gt; 0}:
u(tt) + Delta(2)u + Delta theta = 0 and theta(t) - Delta theta - Delta u(t) = 0 in R-+(n) x (0,infinity),
subject to the boundary condition: u|(xn=0) = D(n)u(|xn=0) = theta|(xn=0) = 0 and initial condition: (u,D(t)u,theta)|(t=0) = (u(0),v(0) theta(0)) is an element of H-p = W-p,D(2) x L-p x L-p, where W-p,D(2) = {u is an element of W-p(2)| u|(xn=0) = D(n)u|(xn=0) = 0}. We show that for any p is an element of (1,infinity), the associated sernigroup {T(t)}(t &gt;= 0) is analytic in the underlying space H-p. Moreover, a solution (u, theta) satisfies the estimates:
parallel to del(j)(del(2)u(.,t),u(t)(.t),theta(.t))parallel to(Lp(R+n)) &lt;= C(p,q)t(-j/2-n/2(1/p-1/q))parallel to(del(2)u(0),v(0),theta(0))parallel to(Lp(R+n)) (t &gt; 0)
for j = 0, 1, 2 provided that 1 &lt; p &lt;= q &lt;= infinity when j = 0, 1 and that 1 &lt; p &lt;= q &lt;= infinity when j = 2, where del(j) stands for space gradient of order j.

We consider the motion of a viscous fluid filling the whole three-dimensional space exterior to a rotating obstacle with constant angular velocity. We develop the L (p) -L (q) estimates and the similar estimates in the Lorentz spaces of the Stokes semigroup with rotation effect. We next apply them to the Navier-Stokes equation to prove the global existence of a unique solution which goes to a stationary flow as t -&gt; a with some definite rates when both the stationary flow and the initial disturbance are sufficiently small in L (3,a) (weak-L (3) space).

The paper is concerned with linear thermoelastic plate equations in a domain Omega:
u(tt) + Delta(2)u + Delta theta = 0 and theta(t) - Delta theta - Delta u(t) = 0 in Omega x (0, infinity),
subject to the Dirichlet boundary condition u|(Gamma) = D(nu)u|(Gamma) = theta|(Gamma) = 0 and initial condition (u, u(t), theta)|(t=0) = (u(0),v(0),theta(0)) is an element of W-p,D(2)(Omega) x L-p x L-p. Here, Omega is a bounded or exterior domain in R-n (n &gt;= 2). We assume that the boundary Gamma of Omega is a C-4 hypersurface and we define W-p,D(2) by the formula W-p,D(2) = {u is an element of W-p(2) : u|(Gamma) = D(nu)u|(Gamma) = 0}. We show that, for any p is an element of (1, infinity), the associated semigroup {T(t)}(t &gt;= 0) is analytic. Moreover, if Omega is bounded, then {T(t)}(t &gt;= 0) is exponentially stable.

In this paper we consider a resolvent problem of the Stokes operator with some boundary condition in the half space, which is obtained as a model problem arising in evolution free boundary problems for viscous, incompressible fluid flow. We show standard resolvent estimates in the L(q) framework (1 &lt; q &lt; infinity), applying some kernel estimates to concrete solution formulas. The Volevich trick in [21] plays a fundamental role in estimating solutions. (C) 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

In this paper, we prove the L-p-L-q maximal regularity of solutions to the Neumann problem for the Stokes equations with non-homogeneous boundary condition and divergence condition in a bounded domain. The result was first stated by Solonnikov [ 17], but he assumed that p = q &gt; 3 and considered only the finite time interval case. In this paper, we consider not only the case: 1 &lt; P, q &lt; infinity but also the infinite time interval case. Especially, we obtain the L-p-L-q maximal regularity theorem with exponential stability on the infinite time interval.

In this paper, we Study a stationary and a nonstationary problem of the Ginzburg-Landau-Maxwell equations with Coulomb gauge in the LP framework. First we prove a unique existence of stationary solution near the constant state with a small external magnetic field. Moreover, we prove a globally in time existence of solutions to the time dependent Ginzburg-Landau-Maxwell equations with simill initial data and external magnetic field, and we show its convergence to the corresponding stationary Solution when time tends to infinity. The key of our approach iS to use various L-p-L-q estimates of the analytic sernigrOUP generated by the linearized problem. Especially our initial data belong to L-3 without any additional regularity. (c) 2005 Published by Elsevier Inc.

We prove a generalized resolvent estimate of Stokes equations with nonhomogeneous Robin boundary condition and divergence condition in the L-q framework (1 &lt; q &lt; infinity) in. a domain of R-n (n &gt;= 2) that is a bounded domain or the exterior of a bounded domain. The Robin condition consists of two conditions: v (.) u = 0 and alpha u + beta(T(u,p)v - &lt; T(u,p)v, v &gt; v) = h on the boundary of the domain with alpha, beta &gt;= 0 and alpha + beta = 1, where u denotes a velocity vector, p a pressure, T(u, p) the stress tensor for the Stokes flow, and v the unit outer normal to the boundary of the domain. It presents the slip condition when 3 = 1 and the non-slip one when a = 1, respectively.

We consider a free boundary problem for the Navier-Stokes equation in R-n (n &gt;= 2). We prove a local in time unique existence theorem for any initial data and a global in time unique existence theorem for some small initial data. The problem we consider in this paper was already treated by V. Solonnikov [15]. But, recently in [10] we proved an L-p-L-q maximal regularity theorem for the Stokes equation with Neumann boundary condition which is a linearized version of the free boundary problem for the Navier-Stokes equation treated in this paper. Our proof is based on this theorem. Therefore our solution is obtained in the space W-q,p(2,1) (2 &lt; p &lt; infinity and n &lt; q &lt; infinity) while a solution in [15] is in W-q(2,1) = W-q,q(2,1) (n &lt; q &lt; infinity) Moreover, our proof of the global in time existence theorem is much simpler than [15], because in [10] we established a maximal regularity theorem on the whole time interval (0, infinity) with exponential stability. The results obtained in this paper were already announced in Shibata-Shimizu [11].

We consider a compressible viscous fluid affected by external forces of general form which are small and smooth enough in suitable norms in R-3. In Shibata and Tanaka [Y. Shibata, K. Tanaka, On the steady flow of compressible viscous fluid and its stability with respect to initial disturbance, J. Math. Soc. Japan 55 (2003) 797-826], we proved the unique existence and some regularity of the steady flow and its globally in-time stability with respect to a small initial disturbance in the H-3 -framework. In this paper, we investigate the rate of the convergence of the non-stationary flow to the corresponding steady flow when the initial data are small enough in the H3 and also belong to L-6/5. (C) 2007 Elsevier Ltd. All rights reserved.

In this paper, we obtain local energy decay estimates and L-p-L-q estimates of the solutions to the Stokes equations with Neumann boundary condition which is obtained as a linearized equation of the free boundary problem for the Navier-Stokes equations. Comparing with the non-slip boundary condition case, we have a better decay estimate for the gradient of the semigroup because of the null force at the boundary.

We prove the L p-L q maximal regularity of solutions to the Neumann problem for the Stokes equations with non-homogeneous boundary condition and divergence condition in a bounded domain. And as an application, we consider a free boundary problem for the Navier-Stokes equation. We prove a locally in time unique existence of solutions to this problem for any initial data and a globally in time unique existence of solutions to this problem for some small initial data.

We discuss L-p-L-q type estimates of the Stokes semigroup and their application to the Navier-Stokes equation in a perturbed half-space. Especially, we have the L-p-L-q type estimate of the gradient of the Stokes semigroup for any p and q with 1 &lt; p &lt;= q &lt; infinity, while the same estimate holds only for the exponents p and q with 1 &lt; p &lt;= q &lt;= n in the exterior domain case, where n denotes the space dimension, and, therefore, we can get better results concerning the asymptotic behaviour of solutions to the Navier-Stokes equations compared with the exterior domain case.
Our proof of the L-p-L-q type estimate of the Stokes semigroup is based on the local energy decay estimate obtained by investigation of the asymptotic behavior of the Stokes resolvent near the origin. The order of asymptotic expansion of the Stokes resolvent near the origin is one half better compared with the exterior domain case, because we have the reflection principle on the boundary in the half-space case unlike the whole space case, and, then, such better asymptotics near the boundary are also obtained in the perturbed half-space by a perturbation argument. This is one of the reasons why the result in the perturbed half-space case is essentially better compared with the exterior domain case.

This paper is concerned with the stationary Navier-Stokes equations in exterior domains of dimension n ≥ 3, and provides a sufficient condition on the external force for the unique solvability. This condition is valid both in the case with small but nonzero velocity at infinity, and in the case with zero velocity at infinity. As a result it is proved that, if the external force satisfies this condition, the solution with nonzero velocity at infinity converges to the solution with zero velocity at infinity with respect to the weak-∗ topology of appropriate function spaces.© 2005, The Mathematical Society of Japan. All rights reserved.

This paper is concerned with the study of the system of Laplace operators with perfect wall condition in the L-p framework. Our study includes a bounded domain, an exterior domain and a domain having noncompact boundary such as a perturbed half space. A direct application of our study is to prove the analyticity of the semigroup corresponding to the Maxwell equation of parabolic type, which appears as a linearized equation in the study of the nonstationary problem concerning the Ginzburg-Landau-Maxwell equation describing the Ginzburg-Landau model for superconductivity, the magnetohydrodynamic equation and the Navier-Stokes equation with Neumann boundary condition. And also, our theory is applicable to some solvability of the stationary problem of these nonlinear equations in the L-p framework.

The linearized initial boundary value problem describing the motion of the viscous compressible fluid is studied under Dirichlet zero condition in bounded and unbounded domains. The resolvent estimate for the corresponding operator is proved in the L, framework and the sharp inner estimate of the resolvent set is obtained. Copyright (C) 2004 John Wiley Sons, Ltd.

In this paper, we prove a local energy decay of the Oseen semigroup in the n-dimensional exterior domain (n greater than or equal to 3). In the three dimensional exterior domain case, Kobayashi and Shlbata [16] already proved the local energy decay. Our theorem is not only an extension of the results due to Kobayashi and Shlbata to the n-dimensional case but also the complete study of the local energy decay theorem for the Oseen equation with optimal time decay rates. The local energy decay gives us a crucial step to obtain the L-p-L-q estimates of the Oseen semigroup, which enable us to prove the unique existence of globally in time solutions to the Navier-Stokes equation in an exterior domain with small initial data in the L-n framework, and their properties of time decay.

We consider the exterior problem in the plane for the wave equation with a Neumann boundary condition and study the asymptotic behavior of the solution for large times. For possible application we are interested to show a decay estimate which does not involve weighted norms of the initial data. In the paper we prove such an estimate, by a combination of the estimate of the local energy decay and decay estimates for the free space solution. (C) 2003 Elsevier Inc. All rights reserved.

We consider a compressible viscous fluid effected by general form external force in R-3. In part 1, an analysis of the linearized problem based on the weighted-L-2 method implies a condition on the external force for the existence and some regularities of the steady flow. In part 2, we study the stability of the steady flow with respect to the initial disturbance. What we proved is: if H-3-norm of the initial disturbance is small enough, then the solution to the non-stationary problem exists uniquely and globally in, time.

Obtained is the L-p estimate of solutions to the resolvent problem for the Stokes system with interface condition in a bounded domain in R-n. It is the first step to consider the free boundary value problem. (C) 2003 Elsevier Science (USA). All rights reserved.

We consider the L-p-L-q estimates of solutions to the Cauchy problem of linearized compressible Navier-Stokes equation. Especially, we investigate the diffusion wave property of the compressible Navier-Stokes flows, which was studied by D. Hoff and K. Zumbrum and Tai- P. Liu and W. Wang.

Tosio Kato's method and principle for evolution equations in mathematical physics (Sapporo, 2001)

We consider the equations of motion of compressible viscous and heat-conductive gases in an exterior domain in R-3. We give the L-q-L-p estimates fur solutions to the linearized equations and show an optimal decay estimate for solutions to the nonlinear problem.

We proved L-q - L-r type estimates of the Stokes semigroup in a two dimensional exterior domain. Our proof is based on the local energy decay estimate obtained by investigation of the asymptotic behavior of the resolvent of the Stokes operator near the origin.

We consider the initial-boundary value problem for a linear thermoelastic plate equation and we prove that the energy associated to the system decays exponentially to zero as time goes to infinity. (C) 1997 by B. G. Teubner Stuttgart-John Wiley & Sons Ltd.

A new approach to the existence theory for the Navier-Stokes equations, using a technique of KATO [15], further developed in combination with estimates for Oseen's equation by KOBAYASHI & SHIBATA [17] and SHIBATA [24], has made possible the solution of a long-standing open problem often referred to as Finn's ''starting problem''. This paper provides the solution.

We prove a global in time existence theorem of classical solutions of the initial boundary value problem For a non-linear thermoviscoelastic equation in a bounded domain for very smooth initial data, external forces and heat supply which are very close to a specific constant equilibrium state. Our proof is a combination of a local in time existence theorem and some a priori estimates of local in time solutions. Such a priori estimates are proved basically for suitable linear problems by using some multiplicative techniques. An exponential stability of the constant equilibrium state also follows from our proof of the existence and regularity theorems.

We prove the global existence of analytic solutions to the non-linear viscoelastic equations of the following type: u(tt) = phi(\\del(x)u\\2)DELTAu + psi(\\D(x)beta1u\\2,...,\\D(x)betaN)\\)DELTAu(t), where phi and psi are continuous and non-negative functions satisfying some additional conditions. The method of the analytic energy estimates is used.

The Neumann problem for the linear thermoelastic plate equations is studied in the bounded domain of any space dimensions. The exponential decay property is investigated as well as the existence theorem.

We are mainly concerned with the Dirichlet initial boundary value problem in one-dimensional nonlinear thermoelasticity. It is proved that if the initial data are close to the equilibrium then the problem admits a unique, global, smooth solution. Moreover, as time tends to infinity, the solution is exponentially stable. As a corollary we also obtain the existence of periodic solutions for small, periodic right-hand sides.

The global existence of smooth solutions to the equations of nonlinear hyperbolic system of 2nd order with third order viscosity is shown for small and smooth initial data in a bounded domain of n-dimensional Euclidean space with smooth boundary. Dirichlet boundary condition is studied and the asymptotic behaviour of exponential decay type of solutions as t tending to infinity is described. Time periodic solutions are also studied. As an application of our main theorem, nonlinear viscoelasticity, strongly damped nonlinear wave equation and acoustic wave equation in viscous conducting fluid are treated.

The global existence of smooth solutions to the equations of nonlinear thermoelasticity is shown for a one-dimensional homogeneous reference configuration. Dirichlet boundary conditions are studied and the asymptotic behaviour of the solutions as t --&gt; infinity is described.